1. Future value Present value Annuities TVM is one of the most important concepts in finance: A dollar today is worth more than a dollar in the future. Why is this true?? How does this affect us?? HW: 2-1 through 2-5, pg 84 –B&E Chapter 2 Time Value of Money

2. Time lines show timing of cash flows. CF 0 CF 1 CF 3 CF 2 0123 i% Tick marks at ends of periods, so Time 0 is today; Time 1 is the end of Period 1; or the beginning of Period 2.

3. Time line for a $100 lump sum due at the end of Year 2. 100 012 Year i%

4. Time line for an ordinary annuity of $100 for 3 years. 100 0123 i%

5. Time line for uneven CFs: -$50 at t = 0 and $100, $75, and $50 at the end of Years 1 through 3. 100 50 75 0123 i% -50

6. What’s the FV of an initial $100 after 3 years if i = 10%? FV = ? 0123 10% Finding FVs (moving to the right on a time line) is called compounding. 100

7. After 1 year: FV 1 = PV + INT 1 = PV + PV (i) = PV(1 + i) = $100(1.10) = $110.00. After 2 years: FV 2 = FV 1 (1+i) = PV(1 + i)(1+i) = PV(1+i) 2 = $100(1.10) 2 = $121.00.

8. After 3 years: FV 3 = FV2(1+i)=PV(1 + i) 2 (1+i) = PV(1+i) 3 = $100(1.10) 3 = $133.10. In general, FV n = PV(1 + i) n.

9. Future Value Relationships

10. Multi-Period Compounding Examples You put $400 into an account that pays 8 % interest compounded annually, quarterly. How much will be in your account in 6 years? Set:P/YR= 1, END, Format ->Dec=4,CLR TVM N=6, I/Y=8, PV=-400 -> FV= 634.75 Interest is compounded 4 times per year, so: 8 % / 4 = 2 % interest rate per period 6 yrs x 4 = 24 periods FV = $643.37 What do you get if you compound daily instead? I//Y=8/365 (not.08!!!!), N=6*365 FV = $646.40

11. Three Ways to Find FVs Solve the equation with a regular calculator. Use a financial calculator. Use a spreadsheet.

12. Financial calculators solve this equation: There are 4 variables. If 3 are known, the calculator will solve for the 4th. Financial Calculator Solution

13. 310-100 0 NI/YR PV PMTFV 133.10 Here’s the setup to find FV: Clearing automatically sets everything to 0, but for safety enter PMT = 0. Set:P/YR = 1, END. INPUTS OUTPUT

14. Spreadsheet Solution Use the FV function: see spreadsheet in Ch 02 Mini Case.xls. – = FV(Rate, Nper, Pmt, PV) – = FV(0.10, 3, 0, -100) = 133.10

15. 10% What’s the PV of $100 due in 3 years if i = 10%? Finding PVs is discounting, and it’s the reverse of compounding. 100 0123 PV = ?

16. Solve FV n = PV(1 + i ) n for PV:  PV= $100 1 1.10 = $1000.7513 = $75.13.       3

17. Financial Calculator Solution 3 10 0100 N I/YR PV PMTFV -75.13 Either PV or FV must be negative. Here PV = -75.13. Put in $75.13 today, take out $100 after 3 years. INPUTS OUTPUT

18. Spreadsheet Solution Use the PV function: see spreadsheet. – = PV(Rate, Nper, Pmt, FV) – = PV(0.10, 3, 0, 100) = -75.13

19. Finding the Time to Double 20% 2 012? FV= PV(1 + i) n $2= $1(1 + 0.20) n (1.2) n = $2/$1 = 2 nLN(1.2)= LN(2) n= LN(2)/LN(1.2) n= 0.693/0.182 = 3.8. e=2.7183, ln(e)=1 10^2=100, LOG(100)=2, Rule of 72 >> 72/periods = IPER

20. 20 -1 0 2 NI/YR PV PMTFV 3.8 INPUTS OUTPUT Financial Calculator

21. Spreadsheet Solution Use the NPER function: see spreadsheet. –= NPER(Rate, Pmt, PV, FV) – = NPER(0.20, 0, -1, 2) = 3.8

22. Finding the Interest Rate ?% 2 0123 FV= PV(1 + i) n $2= $1(1 + i) 3 (2) (1/3) = (1 + i) 1.2599= (1 + i) i= 0.2599 = 25.99%.

23. 3 -1 0 2 NI/YR PV PMTFV 25.99 INPUTS OUTPUT Financial Calculator

24. Spreadsheet Solution Use the RATE function: –= RATE(Nper, Pmt, PV, FV) – = RATE(3, 0, -1, 2) = 0.2599

25. Ordinary Annuity PMT 0123 i% PMT 0123 i% PMT Annuity Due What’s the difference between an ordinary annuity and an annuity due? PVFV

26. What’s the FV of a 3-year ordinary annuity of $100 at 10%? 100 0123 10% 110 121 FV= 331

27. Suppose you can take a penny and double your money every day for 30 days. What will you be worth? Wait!! Guess a value before you calculate. Iper=100%, n=30, pv=-.01, pmt=0, Fv = $10,737,418.24 1cent,2,4,8,16,32,64,128,256,512,1024,2048,4096 cents… Penny :Super TVM Question

28. FV Annuity Formula The future value of an annuity with n periods and an interest rate of i can be found with the following formula:

29. Financial calculators solve this equation: There are 5 variables. If 4 are known, the calculator will solve for the 5th. Financial Calculator Formula for Annuities

30. 3 10 0 -100 331.00 NI/YRPVPMTFV Financial Calculator Solution Have payments but no lump sum PV, so enter 0 for present value. INPUTS OUTPUT

31. Spreadsheet Solution Use the FV function: see spreadsheet. – = FV(Rate, Nper, Pmt, Pv) – = FV(0.10, 3, -100, 0) = 331.00

32. What’s the PV of this ordinary annuity? 100 0123 10% 90.91 82.64 75.13 248.69 = PV, FV=0

33. PV Annuity Formula The present value of an annuity with n periods and an interest rate of i can be found with the following formula:

34. Have payments but no lump sum FV, so enter 0 for future value. 3 10 100 0 NI/YRPVPMTFV -248.69 INPUTS OUTPUT Financial Calculator Solution

35. Spreadsheet Solution Use the PV function: see spreadsheet. – = PV(Rate, Nper, Pmt, Fv) – = PV(0.10, 3, 100, 0) = -248.69

36. Find the FV and PV if the annuity were an annuity due. 100 0123 10% 100

37. PV and FV of Annuity Due vs. Ordinary Annuity PV of annuity due: – = (PV of ordinary annuity) (1+i) –= (248.69) (1+ 0.10) = 273.56 FV of annuity due: –= (FV of ordinary annuity) (1+i) –= (331.00) (1+ 0.10) = 364.1

38. 310 100 0 -273.55 NI/YRPVPMTFV Switch from “End” to “Begin”. Then enter variables to find PVA 3 = $273.55. Then enter PV = 0 and press FV to find FVA 3 = $364.10. INPUTS OUTPUT

39. Excel Function for Annuities Due Change the formula to: =PV(10%,3,-100,0,1) The fourth term, 0, tells the function there are no other cash flows. The fifth term tells the function that it is an annuity due. A similar function gives the future value of an annuity due: =FV(10%,3,-100,0,1)

40. What is the PV of this uneven cash flow stream? 0 100 1 300 2 3 10% -50 4 90.91 247.93 225.39 -34.15 530.08 = PV

41. Input in “CFLO” register: CF 0 = 0 (Typically initial investment so –ve) CF 1 = 100 CF 2 = 300 CF 3 = 300 CF 4 = -50 Enter I = 10%, then press NPV button to get NPV = 530.09. (Here NPV = PV.)

42. Spreadsheet Solution Excel Formula in cell A3: =NPV(10%,B2:E2) ABCDE 101234 2100300300-50 3530.09

43. Simple (Quoted) Rate k SIMPLE = Simple (Quoted) Rate used to compute the interest paid per period Annual Percentage Rate APR = Annual Percentage Rate = k SIMPLE periodic rate X the number of periods per year Effective Annual Rate EAR= Effective Annual Rate the annual rate of interest actually being earned Distinguishing Between Different Interest Rates

44. Nominal rate (i Nom ) Stated in contracts, and quoted by banks and brokers. Not used in calculations or shown on time lines Periods per year (m) must be given. Examples: –8%; Quarterly –8%, Daily interest (365 days)

45. Periodic rate (i Per ) i Per = i Nom /m, where m is number of compounding periods per year. m = 4 for quarterly, 12 for monthly, and 360 or 365 for daily compounding. Used in calculations, shown on time lines. Examples: –8% quarterly: iPer = 8%/4 = 2%. –8% daily (365): iPer = 8%/365 = 0.021918%.

46. Will the FV of a lump sum be larger or smaller if we compound more often, holding the stated I% constant? Why? LARGER! If compounding is more frequent than once a year--for example, semiannually, quarterly, or daily--interest is earned on interest more often.

47. FV Formula with Different Compounding Periods (e.g., $100 at a 12% nominal rate with semiannual compounding for 5 years) = $100(1.06) 10 = $179.08. With annual cmpndg the A=$176.23 FV = PV1.+ i m n Nom mn       FV = $1001+ 0.12 2 5S 2x5      

48. FV of $100 at a 12% nominal rate for 5 years with different compounding FV(Annual)= $100(1.12) 5 = $176.23. FV(Semiannual)= $100(1.06) 10 =$179.08. FV(Quarterly)= $100(1.03) 20 = $180.61. FV(Monthly)= $100(1.01) 60 = $181.67. FV(Daily)= $100(1+(0.12/365)) (5x365) = $182.19.

49. Effective Annual Rate (EAR = EFF%) The EAR is the annual rate which causes PV to grow to the same FV as under multi-period compounding Example: Invest $1 for one year at 12%, semiannual: FV = PV(1 + i Nom /m) m FV = $1 (1.06) 2 = 1.1236. EFF% = 12.36%, because $1 invested for one year at 12% semiannual compounding would grow to the same value as $1 invested for one year at 12.36% annual compounding.

50. An investment with monthly payments is different from one with quarterly payments. Must put on EFF% basis to compare rates of return. Use EFF% only for comparisons. Banks say “interest paid daily.” Same as compounded daily.

51. How do we find EFF% for a nominal rate of 12%, compounded semiannually? EFF% = - 1 ( 1 + ) i Nom m m = - 1.0 ( 1 + ) 0.12 2 2 = (1.06) 2 - 1.0 = 0.1236 = 12.36%.

52. Effective Annual Rate What is the effective annual rate of 12%, compounded monthly? ->( [1+.12/12]^12)-1 = EAR = 12.68% What is the effective annual rate of 12%, compounded daily? -> [1+.12/365]^365-1 = EAR = 12.75% What is m? Number of compounding periods per year. 2 nd ICONV

53. EAR (or EFF%) for a Nominal Rate of of 12% EAR Annual = 12%. EAR Q =(1 + 0.12/4) 4 - 1= 12.55%. EAR M =(1 + 0.12/12) 12 - 1= 12.68%. EAR D(365) =(1 + 0.12/365) 365 - 1= 12.75%.

54. Can the effective rate ever be equal to the nominal rate? Yes, but only if annual compounding is used, i.e., if m = 1. If m > 1, EFF% will always be greater than the nominal rate.

55. When is each rate used? i Nom :Written into contracts, quoted by banks and brokers. Not used in calculations or shown on time lines.

56. i Per :Used in calculations, shown on time lines. If i Nom has annual compounding, then i Per = i Nom /1 = i Nom.

57. (Used for calculations if and only if dealing with annuities where payments don’t match interest compounding periods.) EAR = EFF%: Used to compare returns on investments with different payments per year.

58. Amortization Construct an amortization schedule for a $1,000, 10% annual rate loan with 3 equal payments.

59. Step 1: Find the required payments. PMT 0123 10% -1,000 3 10 -1000 0 INPUTS OUTPUT NI/YRPVFV PMT 402.11

60. Step 2: Find interest charge for Year 1. INT t = Beg bal t (i) INT 1 = $1,000(0.10) = $100. Step 3: Find repayment of principal in Year 1. Repmt = PMT - INT = $402.11 - $100 = $302.11.

61. Step 4: Find ending balance after Year 1. End bal= Beg bal - Repmt = $1,000 - $302.11 = $697.89. Repeat these steps for Years 2 and 3 to complete the amortization table.

62. Interest declines. Tax implications. BEGPRINEND YRBALPMTINTPMTBAL 1$1,000$402$100$302$698 269840270332366 3366402373660 TOT1,206.34206.341,000

63. $ 0123 402.11 Interest 302.11 Level payments. Interest declines because outstanding balance declines. Lender earns 10% on loan outstanding, which is falling. Principal Payments

64. Amortization tables are widely used--for home mortgages, auto loans, business loans, retirement plans, and so on. They are very important! Financial calculators (and spreadsheets) are great for setting up amortization tables.

65. On January 1 (today) you deposit $100 (PV)in an account that pays a nominal interest rate of 11.33463%, with daily compounding (365 days). How much will you have on October 1, or after 9 months (273 days)? (Days given.) Partial number of periods

66. i Per = 11.33463%/365 = 0.031054% per day. FV=? 012273 0.031054% -100 Note: % in calculator, decimal in equation.   FV = $1001.00031054 = $1001.08846= $108.85. 273

67. 273-100 0 108.85 INPUTS OUTPUT N I/YRPVFV PMT i Per =i Nom /m =11.33463/365 =0.031054% per day. Not an Annuity problem Enter i in one step. Leave data in calculator.

68. What’s the value at the end of Year 3 of the following CF stream if the quoted interest rate is 10%, compounded semiannually? 01 100 23 5% 45 6 6-mos. periods 100

69. Payments occur annually, but compounding occurs each 6 months. So we can’t use normal annuity valuation techniques.

70. 1st Method: Compound Each CF 01 100 23 5% 456 100100.00 110.25 121.55 331.80 FVA 3 = $100(1.05) 4 + $100(1.05) 2 + $100 = $331.80.

71. Could you find the FV with a financial calculator? Yes, by following these steps: a. Find the EAR for the quoted rate: 2nd Method: Treat as an Annuity EAR = ( 1 + ) - 1 = 10.25%. 0.10 2 2

72. 3 10.25 0 -100 INPUTS OUTPUT N I/YR PVFV PMT 331.80 b. Use EAR = 10.25% as the annual rate in your calculator: n=3, not 6.

73. What’s the PV of this stream? 0 100 1 5% 23 100 90.70 82.27 74.62 247.59 FV=0, PMT=-100, PV=?

74. You are offered a note which pays $1,000 in 15 months (or 456 days) for $850. You can have $850 in a bank which pays a 6.76649% nominal rate, with 365 daily compounding, which is a daily rate of 0.018538% and an EAR of 7.0%. You plan to leave the money in the bank if you don’t buy the note. The note is riskless. Should you buy it?

75. 3 Ways to Solve: 1. Greatest future wealth: FV 2. Greatest wealth today: PV 3. Highest rate of return: Highest EFF% i Per =0.018538% per day. 1,000 0365456 days -850

76. 1. Greatest Future Wealth Find FV of $850 left in bank for 15 months and compare with note’s FV = $1,000. FV Bank =$850(1.00018538) 456 =$924.97 in bank. Buy the note: $1,000 > $924.97.

77. 456-850 0 924.97 INPUTS OUTPUT NI/YRPVFV PMT Calculator Solution to FV: i Per =i Nom /m =6.76649%/365 =0.018538% per day. Enter i Per in one step.

78. 2. Greatest Present Wealth Find PV of note, and compare with its $850 cost: PV=$1,000/(1.00018538) 456 =$918.95.

79. 456.018538 0 1000 -918.95 INPUTS OUTPUT NI/YRPVFV PMT 6.76649/365 = PV of note is greater than its $850 cost, so buy the note. Raises your wealth.

80. Find the EFF% on note and compare with 7.0% bank pays, which is your opportunity cost of capital: FV n = PV(1 + i) n $1,000 = $850(1 + i) 456 Now we must solve for i. 3. Rate of Return

81. 456-850 0 1000 0.035646% per day INPUTS OUTPUT NI/YRPV FV PMT Convert % to decimal: Decimal = 0.035646/100 = 0.00035646. EAR = EFF%= (1.00035646) 365 - 1 = 13.89%.

82. Using interest conversion ICONV: P/YR=365 NOM%=0.035646(365)= 13.01 EFF%=13.89 Since 13.89% > 7.0% opportunity cost, buy the note.

83. HW 2-6 thru 2-11, 2-20, 2-22, 2-23,